Trees and branches in Banach spaces
HTML articles powered by AMS MathViewer
- by E. Odell and Th. Schlumprecht PDF
- Trans. Amer. Math. Soc. 354 (2002), 4085-4108 Request permission
Abstract:
An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree $\mathcal {T}$ of a certain type on a space $X$ is presumed to have a branch with some property. It is shown that then $X$ can be embedded into a space with an FDD $(E_i)$ so that all normalized sequences in $X$ which are almost a skipped blocking of $(E_i)$ have that property. As an application of our work we prove that if $X$ is a separable reflexive Banach space and for some $1<p<\infty$ and $C<\infty$ every weakly null tree $\mathcal {T}$ on the sphere of $X$ has a branch $C$-equivalent to the unit vector basis of $\ell _p$, then for all $\varepsilon >0$, there exists a subspace of $X$ having finite codimension which $C^2+\varepsilon$ embeds into the $\ell _p$ sum of finite dimensional spaces.References
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- Robert C. James, Uniformly non-square Banach spaces, Ann. of Math. (2) 80 (1964), 542–550. MR 173932, DOI 10.2307/1970663
- W. B. Johnson, On quotients of $L_{p}$ which are quotients of $l_{p}$, Compositio Math. 34 (1977), no. 1, 69–89. MR 454595
- W. B. Johnson and M. Zippin, Subspaces and quotient spaces of $(\sum G_{n})_{l_{p}}$ and $(\sum G_{n})_{c_{0}}$, Israel J. Math. 17 (1974), 50–55. MR 358296, DOI 10.1007/BF02756824
- W. B. Johnson, H. P. Rosenthal, and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488–506. MR 280983, DOI 10.1007/BF02771464
- N.J. Kalton, On subspaces of c$_0$ and extensions of operators into C$(K)$-spaces, Q. J. Math. 52 (2001), 312–328.
- Nigel J. Kalton and Dirk Werner, Property $(M)$, $M$-ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995), 137–178. MR 1324212, DOI 10.1515/crll.1995.461.137
- H. Knaust, E. Odell, and Th. Schlumprecht, On asymptotic structure, the Szlenk index and UKK properties in Banach spaces, Positivity 3 (1999), no. 2, 173–199. MR 1702641, DOI 10.1023/A:1009786603119
- Donald A. Martin, Borel determinacy, Ann. of Math. (2) 102 (1975), no. 2, 363–371. MR 403976, DOI 10.2307/1971035
- B. Maurey, V. D. Milman, and N. Tomczak-Jaegermann, Asymptotic infinite-dimensional theory of Banach spaces, Geometric aspects of functional analysis (Israel, 1992–1994) Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 149–175. MR 1353458
- V. D. Milman, Geometric theory of Banach spaces. II. Geometry of the unit ball, Uspehi Mat. Nauk 26 (1971), no. 6(162), 73–149 (Russian). MR 0420226
- Vitali D. Milman and Nicole Tomczak-Jaegermann, Asymptotic $l_p$ spaces and bounded distortions, Banach spaces (Mérida, 1992) Contemp. Math., vol. 144, Amer. Math. Soc., Providence, RI, 1993, pp. 173–195. MR 1209460, DOI 10.1090/conm/144/1209460
- E. Odell, Applications of Ramsey theorems to Banach space theory, Notes in Banach spaces, Univ. Texas Press, Austin, Tex., 1980, pp. 379–404. MR 606226
- R. Wagner, Finite high-order games and an inductive approach towards Gowers’ dichotomy, Annals of Pure and Applied Logic, 111 (2001), 39–60.
- M. Zippin, Banach spaces with separable duals, Trans. Amer. Math. Soc. 310 (1988), no. 1, 371–379. MR 965758, DOI 10.1090/S0002-9947-1988-0965758-0
Additional Information
- E. Odell
- Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
- Email: odell@math.utexas.edu
- Th. Schlumprecht
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- MR Author ID: 260001
- Email: schlump@math.tamu.edu
- Received by editor(s): October 10, 2000
- Received by editor(s) in revised form: November 7, 2001
- Published electronically: May 20, 2002
- Additional Notes: Research supported by NSF
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 4085-4108
- MSC (2000): Primary 46B03
- DOI: https://doi.org/10.1090/S0002-9947-02-02984-7
- MathSciNet review: 1926866